Aren't all messages on my iPhone and the Message app supposed to be synced? Message app on iMac simply won't show some conversations that are clear on the phone. Some do show up. What's that about? The person I'm messaging is in my contacts which is synced. ?
Make sure you have iCloud Syncing enabled for Messages.
Make sure you have iCloud Syncing enabled for Messages.
iMessages will go to both without the following but if you want regular SMS/MMS text messages to sync, see: How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac - Apple Support
Did you try the following from the previously supplied link and does it work?
Forward older text messages
and enter a recipient. 
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Didn't work. I have one particular person I'm texting today and all is fine on the phone, and nothing shows up on the Message app on my iMac. ALL my other text conversations show up, though.
That's not at all what I'm talking about.
That might have done it. Took a while for the messages to sync. Thanks.
Message app on iMac/Catalina not showing messages that are on my iPhone.
